Question

The shape of the distribution of the time required to get an oil change at a 15-minute oil-change facility is unknown. However, records indicate that the mean time is 16.2 minutes, and the standard deviation is 3.4 minutes. Complete the following:

(a) To compute probabilities regarding the sample mean using the normal model, what size sample would be required?

A. The sample size needs to be less than or equal to 30.
B. The normal model cannot be used if the shape of the distribution is unknown. The sample size needs to be greater than or equal to 30.
C. Any sample size could be used.

(b) What is the probability that a random sample of 35 oil changes results in a sample mean time less than 15 minutes? (Round to four decimal places as needed.)

(c) Suppose the manager agrees to pay each employee a $50 bonus if they meet a certain goal. On a typical Saturday, the oil-change facility will perform 35 oil changes between 10 A.M. and 12 P.M. Treating this as a random sample, there would be a 10% chance of the mean oil-change time being at or below what value? This will be the goal established by the manager. (Round to one decimal place as needed.)

Answer 1

For the oil change distribution problem, a sample size of 30 or more is required to use the normal model for probabilities. To find the probability of a sample mean below a specific time, calculate the SEM and use the Z-score. For a specific success rate, determine the goal by finding the corresponding Z-score and using the sample mean formula involving the population mean and SEM.

Step-by-step explanation:

Understanding Statistical Distributions and Sample Sizes

To answer part (a) of the question regarding the time required to get an oil change, we must consider the Central Limit Theorem (CLT). The CLT states that, regardless of the population distribution shape, the sampling distribution of the mean will approximate a normal distribution if the sample size is sufficiently large. Generally, a sample size of 30 or more is considered enough for the CLT to apply, assuming all samples are independent and identically distributed. Therefore, the correct answer is B. The sample size needs to be greater than or equal to 30.

For part (b), to calculate the probability that a random sample of 35 oil changes results in a sample mean time less than 15 minutes, we use the normal distribution approximation. First, we calculate the standard error of the mean (SEM) as follows: SEM = σ/ sqrt(n), where σ is the population standard deviation, and n is the sample size. Then, we find the Z-score for the sample mean of 15 minutes and use the standard normal table or a calculator to find the corresponding probability.

For part (c), if the manager wants there to be a 10% chance of the mean oil-change time being at or below a certain value, we look at the Z-score corresponding to the 10th percentile in the normal distribution table and use the formula for the sample mean to find the goal for the employees. The formula is: goal = μ + Z * SEM, where μ is the population mean, Z is the Z-score for 10th percentile, and SEM is the standard error of the mean.